[1] We apply rock mechanics concepts to the seismological observations in order to explain why during hydraulic injection some events display tensile and some shear deformation. The presence of non-shear components depends on the differential stress and the fracture orientation with respect to the σ_{1} direction. Provided the slip vector is parallel to the traction we define four types of earthquakes according to the ratio of the shear and tensile components. Assuming a Griffith failure envelope, hybrid events containing both shear and tensile components can occur for fractures striking within 22.5° of σ_{1}. We argue that pure tensile fractures striking parallel to σ_{1} are unlikely in the presence of natural fractures. The low shear traction of tensile events also implies their small stress drops. By applying the analysis to two different data sets, Soultz-sous-Forets and Cotton Valley, we show that different orientations of natural fractures and differential stress in the targeted formations made each region favorable for different non-DC components in the injection-induced seismicity.

[2] The observation of tensile and shear events in microseismic activity is a subject that is currently undergoing large discussions. The change in volume related to tensile events is a proxy to the possible role of high pressure fluids in the seismogenic process. Volume increase indicates an elevated fluid pressure preceding the earthquake, which leads to the coseismic opening of a newly formed or pre-existing fracture. Volume decrease indicates that releasing the fault planes during seismic slip allowed the confining stress to close some of the open space in the fault zone. The information about the shear and volumetric components of the seismic slip is carried by the seismic waves and can be inverted to get the seismic moment tensor (MT). MT contains the information about the orientation of the fault plane, and also about the double-couple (DC) and non-double-couple (non-DC) components of the seismic source which are thought to be signs of fluid presence. For many years the non-DC components were usually neglected, because the data were not of sufficient quality required for a reliable determination of full moment tensors [e.g., Nolen-Hoeksema and Ruff, 2001]. However, few recent studies using high-quality data indicate that the non-DC moment tensors become a more frequent case. Significant non-DC components were found by Šílený et al. [2009] and Baig and Urbancic [2010] in microearthquakes induced by hydraulic injection in gas bearing sediments and by Julian et al. [2010] in injection-induced seismicity in geothermal field. Surprisingly, Horálek et al. [2010] inferred that the seismicity accompanying injections into a granitic geothermal reservoir in Soultz-sous-Forets, France, was pure-DC.

[3] The ambiguous outcome of these studies raises questions regarding the physical and geological conditions for the non-DC components of MT. To address these, we analyze the interplay between stress, fracture orientation and fluid injection to identify the favorable conditions for the occurrence of non-DC earthquakes.

2. Stress and Tensile Earthquakes

[4] The traction on a fault plane is analyzed graphically by Mohr-Coulomb diagrams, which describe the relation between the shear traction τ on the normal traction σ_{n,} resolved on a plane (Figure 1a). In a 2D case the state of stress in the rock is fully described by the Mohr circle of diameter σ_{1} − σ_{3} with the center at (σ_{1} + σ_{3})/2; σ_{1} and σ_{3} being the maximum and minimum effective stress, respectively. The shear and normal tractions on the fracture oriented at the angle θ to the σ_{1} direction are determined by the intersection of the radius at angle 2θ with the Mohr circle. The pressure P of the injected fluid acts against the absolute stress decreasing the effective stresses, which shifts the Mohr circle to lower normal stresses. When the failure strength of the rock is reached, the rock breaks by creating a fresh fracture oriented according to the internal rock friction μ_{i}. The resulting non-linear strength envelope is often replaced by a linearized Mohr-Coulomb failure criterion (Figure 1a)

which proves valid for most rocks in the compressive domain [e.g., Zoback, 2007]; here S_{0} is the rock cohesion.

[5] In order to learn about the behaviour of rocks in the tensile regime one must focus on the relatively small and strongly non-linear region of the strength envelope. In terms of possible coseismic volume changes the intersection of Mohr circle with the strength envelope facilitates the definition of stress intervals of tensile (σ_{n} < 0, τ_{f} = 0), hybrid tensile (σ_{n} < 0, ∣τ_{f}∣ > 0), pure shear (σ_{n} = 0, ∣τ_{f}∣ > 0) and shear (σ_{n} < 0, ∣τ_{f}∣ > 0) fractures (Figure 1b). The hybrid tensile fractures, which represent a continuous transition from tensile to pure shear fractures are also known in the literature as combined fractures, or oblique-tensile fractures. Their existence has been questioned based both on observations in the field and linear fracture-mechanics models. However, strong support for their occurrence was shown by the laboratory experiments of Ramsey and Chester [2004]. This is also consistent with the model for tensile earthquakes, where the opening of the fault is represented by a slip vector that deviates from the fault plane [Vavryčuk, 2001]. Provided the slip takes place along the traction, the tensile, hybrid tensile, pure shear and shear fractures correspond to the slip vector oriented perpendicular, inclined inward, parallel and inclined outward to the fault plane, respectively (Figure 1b). The latter case occurs however rarely, because of missing open space in the fault zone.

[6] The probability of the non-shear component depends on the differential stress represented by the diameter of the Mohr circle so that the Mohr circle intersects the strength envelope at the point where effective normal stress is negative and shear stress is smaller than cohesion. The condition for the tensile component thus depends on the failure envelope and fracture orientation θ. To derive the relation among the tractions τ_{f} and σ_{n}, fracture orientation θ and differential stress (σ_{1} − σ_{3}) we aim to find the coordinates of the intersection of the failure envelope and Mohr circle. We assume a Griffith failure criterion τ_{f}^{2} = S_{0}(2σ_{n} + S_{0}) and employ the equality of derivatives δτ_{f}/δσ_{n} of the parabolic failure criterion and Mohr circle at the point of their intersection and the fact that δτ_{f}/δσ_{n} = cotan(2θ). We get

and

The parabolic Griffith failure criterion then gives

The analysis of the dependence of the shear (equation (3)), normal (equation (5)) and differential (equation (4)) stresses on the fracture orientation θ shows that differential stress at which tensile opening occurs must be smaller than 2S_{0}. The maximum angle of fractures θ_{MAX} and of the differential stress (σ_{1} − σ_{3})_{MAX} at which the tensile opening occurs is obtained for σ_{n} = 0, i.e., τ_{f} = S_{0} (Figure 1b). This gives θ_{MAX} = 22.5° and (σ_{1} − σ_{3})_{MAX} = 2S_{0}. In other words, for the Griffith failure criterion the tensile opening will occur on all fractures oriented within 22.5° of SHmax if the differential stress is smaller than ∼2.8 S_{0.} If one these two conditions is not met, the rock will fail in shear mode. For the typical rock cohesion S_{0} = 20 MPa the maximum differential stress for the occurrence of tensile faulting thus reaches 56 MPa. Under real conditions this value is probably much smaller due to preexisting fractures with significantly smaller cohesion.

[7] The differential stress increases with depth [Zoback, 2007], which could result in a decreasing occurrence of crack opening with depth. However, in overpressured formations where pore pressure approaches the vertical stress the least principal stress must always exceed the pore pressure [Zoback, 2007] in order to avoid natural hydrofracturing. Thus all three principal stresses must be close to the vertical stress, which puts significant constraints on the differential stress. The gradual overpressure buildup in the geological past resulted in a stepwise release of differential stress on well-oriented faults (Figure 1b) [Zoback, 2007]. As a result, the increase in pore pressure leads to smaller differential stresses and a higher probability of crack opening in overpressurized formations.

[8] It is rather unlikely that a tensile fracture exactly parallel to σ_{1} would be created because any fracture oriented close to σ_{1} has a smaller cohesion than the intact rock and will rupture as hybrid tensile prior to creating a tensile fracture with zero shear component (see Figure 1c). The deviation of natural fractures from σ_{1} direction is probably also the reason for the departure of hydraulic fractures from the σ_{1} direction, which can reach up to 40° [Miller et al., 1994] because the injected fluid preferentially stimulates the preexisting fractures.

3. Application to Data

[9] We test the prediction that tensile earthquakes occur on fractures oriented close to the σ_{1} (SHmax) direction if the differential stress is small using two examples of injection induced seismicity.

3.1. Cotton Valley

[10] The hydraulic stimulation of gas-bearing sands in the Cotton Valley formation in Texas [Rutledge et al., 2004] was carried out at the depths of 2650 (treatments A and C) and 2800 m (treatments B and E). Precise locations of the induced microearthquakes showed narrow bands of seismicity occurring along vertical fractures trending within 5° to SHmax in both depth intervals, and fracture sets trending within 20–30° to SHmax for the deeper treatments B and E. The source mechanisms showed strike-slip induced on natural fractures that strike along the bands of seismicity. The treatments A and C comprise both left- and right-lateral strike-slip with strike angles differing by 10° from each other.

[11] The stress data available [Rosepier, 1979; Mayerhofer et al., 2000] for the Cotton Valley sandstone constrain the lithostatic stress (Table 1) and indicate that the tectonic regime is extensional. Although the overlying formation was overpressured in the past [Laubach, 1988], no data on the formation pressure of the Cotton Valley sandstone are available. The tensile strength of the sandstone is larger than 2.3 MPa. The fact that the frictional rock strength constrains the ratio of the maximum and minimum effective stress enables us to determine the friction coefficient using the equation = [(μ^{2} + 1)^{1/2} + μ]^{2} [Zoback, 2007]. By varying the formation pressure P (Table 1) in order to get the same coefficient of friction at both depths, we get μ = 1.0, which leads to a slight overpressure for the deeper treatments. A failure envelope for the Cotton Valley treatments in Figure 2a is based on the choice of μ = 1.0 and S_{0} = 9 MPa (note that S_{0} = 2T_{0} for the Griffith envelope). It is very likely that poroelastic effects would add a few MPa of compressional stress and therefore move the Mohr circle to higher normal-stress values reducing the upper limit of cohesion that allows for tensile components down to S_{0} ∼ 5 MPa.

Table 1. Stress Azimuth and Magnitudes for the Soultz and Cotton Valley^{a}

[12] Because the fault planes initiated during treatments A, B, C, and E strike less than 10° off SHmax, which is well within the maximum angle of 22.5° delineating the extensional failure mode (Figure 2a), the effective normal traction must be negative with σ_{n}/S_{0} < −0.485 (equation (5)) and tensile faulting should occur. During treatments B and E, simultaneously a second set of natural fractures was activated with fault planes striking about 30° off SHmax, which is consistent with failure in the shear mode, see Figure 2a. Shear stress and therefore the potential for the stress drop is smaller on the fractures with small angles (<10°) to SHmax than on the optimally oriented fractures (30°) (Figure 2). This could explain why the hybrid tensile events show smaller seismic moments than the shear events (e.g., cluster 4 of Rutledge et al. [2004]).

3.2. Soultz-sous-Forets

[13] The hydraulic treatments of the geothermal field in Soultz-sous-Forets in the 2003-experiment stimulated natural fractures in the depth range from 3.8 to 5.4 km. The moment tensors of 45 selected earthquakes [Horálek et al., 2010] showed only negligible non-DC components; their focal mechanisms are shown in Figure 2b. The stress analysis (Table 1) gives hydrostatic formation pressure with a stable orientation of the minimum stress σ_{3} of 260° and rotation of the maximum stress σ_{1} from a subvertical to a horizontal orientation. We used the focal mechanisms [Horálek et al., 2010] and the minimum stress orientation (Table 1) to calculate the angle θ and construct the Mohr-Coulomb diagram for the stimulated fractures (Figure 2b). We used the NS oriented nodal planes, which are more optimally oriented for faulting (mean angle 41°) than the other nodal planes (mean angle 55°). The diagram shows that all the microearthquakes failed in the shear mode, due to the fact that both the fault angles and the differential stress lie beyond the limits plausible for hybrid tensile failure. The inferred shear faulting agrees well with the pure-DC character of the moment tensors.

4. Discussion and Conclusions

[14] Seismological studies of fluid injections over the last decade were focused on the detection of tensile events. Their rarity raised the question of whether these events are so small that they are not detectable or whether they even exist. To explore the conditions for their occurrence we combined the approaches of seismology and rock mechanics. Our analysis is based on the Griffith failure criterion whose simple analytic form enables determination of the mutual interdependence of fault traction components, fracture orientation and differential stress that result in different modes of failure. We analyzed the static stress that precedes the earthquake nucleation. Hence, our results don't say anything about the dynamic effects taking place during fracture propagation and dynamically generated tensile opening. We justify our approach by the fact that many injection-induced earthquakes show significant non-DC components.

[15] We find that tensile events would be limited to fractures oriented close to the SHmax direction and regions with differential stress that is small compared to the rock cohesion. These conditions involve either shallow depths (few kms) at hydrostatic conditions or overpressurized reservoirs at greater depths. By applying the analysis to two different data sets, Soultz-sous-Forets and Cotton Valley, we show that different orientations of natural fractures and differential stress in the targeted formations made each region favorable for different non-DC components in the injection-induced seismicity. The smaller resolved shear stress on the hybrid fractures than on the shear fractures can be responsible for overall smaller magnitudes of these events, which also would make them more difficult to detect and be chosen for determining the seismic moment tensor.

Acknowledgments

[16] We thank B. R. Julian, M. D. Zoback, one anonymous reviewer, and R. Harris for comments and suggestions that improved our manuscript. The work of TF was supported by the Czech research projects MSM0021620855 and IAA300120905.